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Homework for 2/26 Due 3/12
1. The desired percentage of SiO2 in a certain type of aluminous cement is
5.5. To test whether the true average percentage is 5.5 for a particular
production facility, 16 independently obtained samples are analyzed. The
sample mean is x̄ = 5.25 and the sample standard deviation is s = 0.28
Suppose that the percentage of SiO2 in a sample is normally distributed.
a. Does this indicate conclusively that the true average percentage differs from 5.5, at significance level α = 0.01?
b. If the true average percentage is µ = 5.6 and a level α = 0.01 test
based on n = 16 is used, what is the probability of detecting this
departure from H0 , that is, the power of the test? (Assume the
population standard deviation is σ = 0.3.)
2. The article “Uncertainty Estimation in Railway Track Life-Cycle Cost”
(J. of Rail and Rapid Transit, 2009) presented the following data on time
to repair (min) a rail break in the high rail on a curved track of a certain
railway line.
159
120
480
149
270
547
340
43
228
202
240
218
A normal probability plot of the data shows a reasonably linear pattern,
so it is plausible that the population distribution of repair time is at least
approximately normal. The sample mean and standard deviation are 249.7
and 145.1, respectively.
a. Is there compelling evidence for concluding that true average repair
time exceeds 200 min? Carry out a test of hypotheses using a significance level of .05.
b. Using σ = 150, what is the type II error probability of the test used
in (a) when true average repair time is actually 300 min? That is,
what is β(300)?
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